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Integral Help

  1. Dec 6, 2007 #1
    1. The problem statement, all variables and given/known data


    I've been thinking over this for the past few days...I'm still stuck though
    Can this integral even be expressed with elementary functions?

    2. Relevant equations

    3. The attempt at a solution
    use the substitution [tex]u=tan{x}[/tex], and then, use integration by parts.
    However I end up with [tex]\int\left|ln(cosx)\right|[/tex], as a term, which I cannot manage to integrate.
    Last edited: Dec 6, 2007
  2. jcsd
  3. Dec 6, 2007 #2
    Hint: Use u=arctan(x). What is du?
  4. Dec 7, 2007 #3
    I think this integral does not have a "classical" primitive. After the substitution and partial integration you end up with:

    [tex]I=\frac{1}{2}\frac{arctan^2(x)}{x^2}+\frac{1}{2}\int \frac{u^2}{sin^2(u)}du[/tex]

    The remaining integral is not an elementary function, according to "the integrator" of mathematica.

    @silver-rose: What is expected as a result? A classical function or an advanced one?
  5. Dec 7, 2007 #4


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    I agree. I can't find an elementary function despite a few pages of calculations and random substitutions. Maybe it's because I'm dumb or something. Anyone else had better luck here?

    Tried it out at http://integrals.wolfram.com/

    The answer was given in some weird notation involving something called a polylogarithm. What's that?
    Last edited: Dec 7, 2007
  6. Dec 7, 2007 #5
    Sorry my bad, I thought the OP had got it wrong in substituting u=tan(x) instead of arctan(x), and didn't check further. Since the mathematica integrator doesn't report a solution in terms of elementary functions, it is highly unlikely that there actually exists one.
  7. Dec 9, 2007 #6
    yea that's what i think so too..

    I've spent days on this integral, basically trying tons and tons of substitutions.

    ti-89 can't do it, and mathematica gives a non-elementary answer.

    Thanks anyways guys.
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